Governed Adaptation: A Law-Level Framework for Long-Horizon Safety in Self-Updating Systems

Adaptive and self-updating systems are increasingly deployed in open-ended and long-horizon environments, yet prevailing safety and alignment approaches rely primarily on empirical training, optimisation heuristics, or architecture-specific controls. This release presents a unified, law-level framework for long-horizon safety in adaptive systems, grounded in invariant-based governance rather than alignment or optimisation. We first introduce a minimal governance law showing that when system updates are subject to explicit admissibility gates enforcing invariant preservation including Lyapunov stability and non-increasing harmsafety guarantees hold by construction, independent of learning method, objective function, or scale. We then prove a negative no-free-lunch theorem establishing that, in the absence of such gates, unguided adaptive systems inevitably admit trajectories of increasing harm over arbitrary horizons, rendering universal safety guarantees impossible in principle. Finally, we unify these results through Codex V∞, a functorial law that lifts governed adaptation to the infinite horizon. Codex V∞ maps adaptive systems to invariant outcome classes that are independent of implementation details and yields a binary criterion for long-term safety: systems either admit a well-defined V∞ object or are ill-defined over infinite horizon. Together, these results establish governance not alignment, training, or scale as a necessary and sufficient condition for long-horizon safety in adaptive systems.